R, the software environment for statistical computing and graphics, supports a wide variety of probability distributions. One of these distributions is the Gamma Distribution, which is particularly useful for modeling the amount of time until an event occurs. In this article, we’ll explore the Gamma Distribution in detail, diving into its definitions, properties, and applications, and then look at how to work with it in R.
Introduction to the Gamma Distribution
The Gamma Distribution is a continuous probability distribution that has two parameters, namely the shape parameter (k) and the scale parameter (θ). It is often used to model the amount of time until an event occurs, such as the time between telephone calls arriving in a call center or the life lengths of electronic components. Its probability density function is given by:
f(x; k,θ) = (x^(k-1)*e^(-x/θ)) / (θ^k*Γ(k))where Γ(k) is the gamma function.
There are two special cases of the Gamma Distribution: the Exponential Distribution (when k=1) and the Chi-Square Distribution (when k=n/2 and θ=2).
Basic Functions in R for the Gamma Distribution
R provides several built-in functions for the Gamma Distribution:
dgamma(x, shape, scale): This function calculates the density of the gamma distribution given a vector of quantiles (x), the shape, and the scale.pgamma(q, shape, scale): This function calculates the cumulative distribution function of the gamma distribution given a vector of quantiles (q), the shape, and the scale.qgamma(p, shape, scale): This function calculates the quantile function (inverse cumulative distribution function) of the gamma distribution given a vector of probabilities (p), the shape, and the scale.rgamma(n, shape, scale): This function generates random variates from the gamma distribution given a number of observations (n), the shape, and the scale.
Let’s dive into how to use these functions.
Working with the Gamma Distribution in R
Density of the Gamma Distribution
We’ll begin by using dgamma() to calculate the density of the Gamma Distribution. Here’s an example:
# Set parameters
shape <- 2
scale <- 2
# Create a sequence of x values
x <- seq(0, 10, 0.01)
# Calculate the density
density <- dgamma(x, shape, scale)
# Plot the density
plot(x, density, type="l", main="Gamma Distribution Density", xlab="x", ylab="Density")
This code calculates the density of a Gamma Distribution with shape=2 and scale=2 at a range of x values from 0 to 10, and then plots the density.
Cumulative Distribution Function of the Gamma Distribution
Next, we’ll use pgamma() to calculate the cumulative distribution function of the Gamma Distribution. Here’s an example:
# Set parameters
shape <- 2
scale <- 2
# Create a sequence of x values
x <- seq(0, 10, 0.01)
# Calculate the cumulative distribution function
cdf <- pgamma(x, shape, scale)
# Plot the cumulative distribution function
plot(x, cdf, type="l", main="Gamma Distribution CDF", xlab="x", ylab="CDF")
This code calculates the cumulative distribution function of a Gamma Distribution with shape=2 and scale=2 at a range of x values from 0 to 10, and then plots the CDF.
Quantile Function of the Gamma Distribution
We’ll now use qgamma() to calculate the quantile function of the Gamma Distribution. Here’s an example:
# Set parameters
shape <- 2
scale <- 2
# Create a sequence of p values
p <- seq(0, 1, 0.01)
# Calculate the quantile function
quantiles <- qgamma(p, shape, scale)
# Plot the quantile function
plot(p, quantiles, type="l", main="Gamma Distribution Quantile Function", xlab="p", ylab="Quantile")
This code calculates the quantile function of a Gamma Distribution with shape=2 and scale=2 for a range of probabilities from 0 to 1, and then plots the quantile function.
Generating Random Variates from the Gamma Distribution
Lastly, we’ll use rgamma() to generate random variates from the Gamma Distribution. Here’s an example:
# Set parameters
shape <- 2
scale <- 2
n <- 1000
# Generate random variates
random_variates <- rgamma(n, shape, scale)
# Plot the histogram of the random variates
hist(random_variates, main="Histogram of Random Variates from Gamma Distribution", xlab="Value", ylab="Frequency")
This code generates 1000 random variates from a Gamma Distribution with shape=2 and scale=2, and then plots a histogram of the variates.
Estimating the Parameters of a Gamma Distribution
We often have data that we believe follows a Gamma Distribution and we want to estimate the shape and scale parameters from this data. This can be done with the fitdistr() function from the MASS package in R. Here’s an example:
# Load the MASS package
library(MASS)
# Generate some data
data <- rgamma(1000, shape=2, scale=2)
# Fit a gamma distribution to the data
fit <- fitdistr(data, "gamma")
# Print the fit
print(fit)This code generates 1000 data points from a Gamma Distribution with shape=2 and scale=2, fits a Gamma Distribution to the data, and then prints the estimated shape and scale parameters.
Conclusion
The Gamma Distribution is a powerful tool for modeling the amount of time until an event occurs. In this article, we covered its definitions, properties, and applications, and looked at how to work with it in R. With R’s built-in functions, it is straightforward to calculate the density, cumulative distribution function, quantile function, and generate random variates from the Gamma Distribution. And with the fitdistr() function from the MASS package, we can even estimate the parameters of a Gamma Distribution from data.